Consider the following graphs of linear functions, all of which have the same scales on the two axes. List the graphs in order of increasing rate of change of 𝑦 with respect to 𝑥.
The rate of change is another way of saying the slope of the graph or its gradient. Which of these photographs have a negative gradient? Well, a negative gradient slopes downwards from left to right. So graph (b) and graph (c) both have a negative gradient or slope. This means that they have a lower rate of change than the two with a positive slope.
The graph with the steepest negative slope will have the least rate of change. Therefore, we can say that graph (b) has the lowest rate of change. This is followed by graph (c) as it is the only other graph with a negative slope or rate of change.
This leaves us with our two positive slopes, graph (a) and graph (d). Which of these two lines is the steeper? As graph (a) is steeper than graph (d), we can say that graph (a) will have the highest rate of change of 𝑦 with respect to 𝑥. This means that the order of the graphs in increasing rate of change of 𝑦 with respect to 𝑥 is (b) then (c) then (d) and then (a). The greater the slope, the greater the rate of change.